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Let $P$ be a poset, partially ordered by $\leq$. An element $a\in P$ is called an *atom* if it covers some minimal element of $P$. As a result, an atom is never minimal. A poset $P$ is called *atomic* if for every element $p\in P$ that is not minimal has an atom $a$ such that $a\leq p$.

Examples.

1. 2. $\mathbb{Z}^{+}$ is partially ordered if we define $a\leq b$ to mean that $a\mid b$. Then $1$ is a minimal element and any prime number $p$ is an atom.

Remark. Given a lattice $L$ with underlying poset $P$, an element $a\in L$ is called an *atom* (of $L$) if it is an atom in $P$. A lattice is a called an *atomic lattice* if its underlying poset is atomic. An *atomistic lattice* is an atomic lattice such that each element that is not minimal is a join of atoms. If $a$ is an atom in a semimodular lattice $L$, and if $a$ is not under $x$, then $a\vee x$ is an atom in any interval lattice $I$ where $x=\bigwedge I$.

Examples.

1. $P=2^{A}$, with the usual intersection and union as the lattice operations meet and join, is atomistic: every subset $B$ of $A$ is the union of all the singleton subsets of $B$.

2. $\mathbb{Z}^{+}$, partially ordered as above, with lattice binary operations defined by $a\wedge b=\operatorname{gcd}(a,b)$, and $a\vee b=\operatorname{lcm}(a,b)$, is a lattice that is atomic, as we have seen earlier. But it is not atomistic: $4$ is not a join of $2$’s; $36$ is not a join of $2$ and $3$ are just two counterexamples.

## Mathematics Subject Classification

06A06*no label found*06B99

*no label found*

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## Comments

## Point lattices

It seems that the term "point lattice" is sometimes used in the same sense as "atomistic lattice" as defined here. Should put a note about this terminology.

Victor Porton - http://www.mathematics21.org

* Algebraic General Topology and Math Synthesis

* 21 Century Math Method (post axiomatic math logic)

* Category Theory - new concepts

## Re: Point lattices

Can you give me some references as to where "point lattice" is used?

## Re: Point lattices

I have found a Russian term in Russian book.

An online dictionary has translated this term

as ``point lattice''. (I assume that the online

dictionary http://lingvo.yandex.ru knows what

it says because it has a translation for

whole two-words term, not ``point'' separately

and ``lattice'' separately.)

It seems that ``point lattice'' is the same

as ``atomistic lattice''. Both terms are rare.

What's our choice?

Victor Porton - http://www.mathematics21.org

* Algebraic General Topology and Math Synthesis

* 21 Century Math Method (post axiomatic math logic)

* Category Theory - new concepts

## Re: Point lattices

> It seems that ``point lattice'' is the same

> as ``atomistic lattice''. Both terms are rare.

> What's our choice?

Unfortunately, using an online dictionary is not an

optimal way to translate mathematical terminology.

The term ``atomistic lattice'' is defined in Gr\"atzer's book

(General Lattice Theory). For a fairly recent example of an

article using the term, consult:

F. Wehrung, A uniform refinement property for congruence

lattices, Proc. Amer. Math. Soc. 127 (1999), 363--370.

The only references to ``point lattices'' I've been able

to find so far are talking about the kind of lattices that

consist of regularly-spaced points in some Euclidean space.